So I’ve finally managed to bang my dissertation into something more or less ready for public consumption. It is basically finished (except for some typos and spell checking).
It is available on my new website.
The title is “The Classification of Two-Dimensional Extended Topological Field Theories”.
I will continue to provide updates and improvements as they happen.
We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological field theories with arbitrary target bicategory. As an immediate corollary we obtain a concrete classification when the target is the symmetric monoidal bicategory of algebras, bimodules, and intertwiners over a fixed commutative ground ring. In the oriented case, such an extended topological field theory is equivalent to specifying a separable symmetric Frobenius algebra.
Along the way we collect together the notion of symmetric monoidal bicategory and define a precise notion of generators and relations for symmetric monoidal bicategories. Given generators and relations, we prove an abstract existence theorem for a symmetric monoidal bicategory which satisfies a universal property with respect to this data. We also prove a theorem which provides a simple list of criteria for determining when a morphism of symmetric monoidal bicategories is an equivalence. We introduce the symmetric monoidal bicategory of bordisms with structure, where the allowed structures are essentially any structures which have a suitable sheaf or stack gluing property.
We modify the techniques used in the proof of Cerf theory and the classification of small codimension singularities to obtain a bicategorical decomposition theorem for surfaces. Moreover these techniques produce a finite list of local relations which are sufficient to pass between any two decompositions. We deliberately avoid the use of the classification of surfaces, and consequently our techniques are readily adaptable to higher dimensions. Although constructed for the unoriented case, our decomposition theorem is engineered to generalize to the case of bordisms with structure. We demonstrate this for the case of bordisms with orientations, which leads to a similar classification theorem.